Bounds for Positions cut : If and are out of bounds, then so is . A position that is out of bounds is overcommitted . Greg Restall Defining Rules, Proofs and Counterexamples 12 of 43 ▶ identity : [ A : A ] is out of bounds. ▶ weakening : If [ X : Y ] is out of bounds, then [ X, A : Y ] and [ X : A, Y ] are also out of bounds.
Bounds for Positions A position that is out of bounds is overcommitted . Greg Restall Defining Rules, Proofs and Counterexamples 12 of 43 ▶ identity : [ A : A ] is out of bounds. ▶ weakening : If [ X : Y ] is out of bounds, then [ X, A : Y ] and [ X : A, Y ] are also out of bounds. ▶ cut : If [ X, A : Y ] and [ X : A, Y ] are out of bounds, then so is [ X : Y ] .
Greg Restall Bounds for Positions Defining Rules, Proofs and Counterexamples 12 of 43 ▶ identity : [ A : A ] is out of bounds. ▶ weakening : If [ X : Y ] is out of bounds, then [ X, A : Y ] and [ X : A, Y ] are also out of bounds. ▶ cut : If [ X, A : Y ] and [ X : A, Y ] are out of bounds, then so is [ X : Y ] . ▶ A position that is out of bounds is overcommitted .
On Cut Suppose is not out of bounds. Suppose is out of bounds. Ask the question: ? The answer no is forced, as a yes answer is excluded (given our other commitments). Greg Restall Defining Rules, Proofs and Counterexamples 13 of 43
On Cut Suppose is out of bounds. Ask the question: ? The answer no is forced, as a yes answer is excluded (given our other commitments). Greg Restall Defining Rules, Proofs and Counterexamples 13 of 43 Suppose [ X : Y ] is not out of bounds.
On Cut Ask the question: ? The answer no is forced, as a yes answer is excluded (given our other commitments). Greg Restall Defining Rules, Proofs and Counterexamples 13 of 43 Suppose [ X : Y ] is not out of bounds. Suppose [ X, A : Y ] is out of bounds.
On Cut The answer no is forced, as a yes answer is excluded (given our other commitments). Greg Restall Defining Rules, Proofs and Counterexamples 13 of 43 Suppose [ X : Y ] is not out of bounds. Suppose [ X, A : Y ] is out of bounds. Ask the question: A ?
On Cut The answer no is forced, as a yes answer is excluded (given our other commitments). Greg Restall Defining Rules, Proofs and Counterexamples 13 of 43 Suppose [ X : Y ] is not out of bounds. Suppose [ X, A : Y ] is out of bounds. Ask the question: A ?
Structural Rules K Defining Rules, Proofs and Counterexamples Greg Restall W W K Cut 14 of 43 X ′ , A � Y ′ X � A, Y A � A Id X, X ′ � Y, Y ′ X � Y X � Y X � A, Y X, A � Y X � A, A, Y X, A, A � Y X � A, Y X, A � Y
The Power of Bounds: Comparatives strong transitivity : weak transitivity : strong irreflexivity : weak reflexivity : contraries : subcontraries : strength : preservation : Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43 Fs s > F t s ⩾ F t
The Power of Bounds: Comparatives strong transitivity : weak transitivity : strong irreflexivity : weak reflexivity : contraries : subcontraries : strength : preservation : Greg Restall Defining Rules, Proofs and Counterexamples 15 of 43 Fs s > F t s ⩾ F t s > F t, t > F u � s > F u
The Power of Bounds: Comparatives weak reflexivity : Defining Rules, Proofs and Counterexamples Greg Restall preservation : strength : subcontraries : contraries : strong irreflexivity : weak transitivity : strong transitivity : 15 of 43 Fs s > F t s ⩾ F t s > F t, t > F u � s > F u s ⩾ F t, t ⩾ F u � s ⩾ F u
The Power of Bounds: Comparatives weak reflexivity : Defining Rules, Proofs and Counterexamples Greg Restall preservation : strength : subcontraries : contraries : 15 of 43 strong irreflexivity : weak transitivity : strong transitivity : Fs s > F t s ⩾ F t s > F t, t > F u � s > F u s ⩾ F t, t ⩾ F u � s ⩾ F u s > F s �
The Power of Bounds: Comparatives weak reflexivity : Defining Rules, Proofs and Counterexamples Greg Restall preservation : strength : subcontraries : contraries : 15 of 43 strong irreflexivity : weak transitivity : strong transitivity : Fs s > F t s ⩾ F t s > F t, t > F u � s > F u s ⩾ F t, t ⩾ F u � s ⩾ F u s > F s � � s ⩾ F s
The Power of Bounds: Comparatives contraries : Defining Rules, Proofs and Counterexamples Greg Restall preservation : strength : subcontraries : weak reflexivity : weak transitivity : strong irreflexivity : strong transitivity : 15 of 43 Fs s > F t s ⩾ F t s > F t, t > F u � s > F u s ⩾ F t, t ⩾ F u � s ⩾ F u s > F s � � s ⩾ F s s > F t, t ⩾ F s �
The Power of Bounds: Comparatives contraries : Defining Rules, Proofs and Counterexamples Greg Restall preservation : strength : subcontraries : weak reflexivity : weak transitivity : strong irreflexivity : strong transitivity : 15 of 43 Fs s > F t s ⩾ F t s > F t, t > F u � s > F u s ⩾ F t, t ⩾ F u � s ⩾ F u s > F s � � s ⩾ F s s > F t, t ⩾ F s � � s > F t, t ⩾ F s
The Power of Bounds: Comparatives weak reflexivity : Defining Rules, Proofs and Counterexamples Greg Restall preservation : strength : subcontraries : contraries : 15 of 43 strong irreflexivity : weak transitivity : strong transitivity : Fs s > F t s ⩾ F t s > F t, t > F u � s > F u s ⩾ F t, t ⩾ F u � s ⩾ F u s > F s � � s ⩾ F s s > F t, t ⩾ F s � � s > F t, t ⩾ F s s > F t � s ⩾ F t
The Power of Bounds: Comparatives weak reflexivity : Defining Rules, Proofs and Counterexamples Greg Restall preservation : strength : subcontraries : contraries : 15 of 43 strong irreflexivity : weak transitivity : strong transitivity : Fs s > F t s ⩾ F t s > F t, t > F u � s > F u s ⩾ F t, t ⩾ F u � s ⩾ F u s > F s � � s ⩾ F s s > F t, t ⩾ F s � � s > F t, t ⩾ F s s > F t � s ⩾ F t Fs, t ⩾ F s � Ft
definitions
How do you define a concept? By showing people how to use it. Greg Restall Defining Rules, Proofs and Counterexamples 17 of 43
Explicit Definition Define a concept by showing how you can compose that concept out of more primitive concepts. isasquare df isarectangle all sidesof are equal inlength . Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions. Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43
Explicit Definition Define a concept by showing how you can compose that concept out of more primitive concepts. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions. Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43 = df x isarectangle ∧ x isasquare all sidesof x are equal inlength .
Explicit Definition Define a concept by showing how you can compose that concept out of more primitive concepts. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions. Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43 = df x isarectangle ∧ x isasquare all sidesof x are equal inlength .
Explicit Definition Define a concept by showing how you can compose that concept out of more primitive concepts. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions. Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43 = df x isarectangle ∧ x isasquare all sidesof x are equal inlength .
Explicit Definition Define a concept by showing how you can compose that concept out of more primitive concepts. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions. Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43 = df x isarectangle ∧ x isasquare all sidesof x are equal inlength .
Explicit Definition Define a concept by showing how you can compose that concept out of more primitive concepts. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions. Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43 = df x isarectangle ∧ x isasquare all sidesof x are equal inlength .
Explicit Definition Define a concept by showing how you can compose that concept out of more primitive concepts. Concepts defined explicitly are sharply delimited (contingent on people accepting the definition). Logical concepts are similarly sharply delimited, but they cannot all be given explicit definitions. Greg Restall Defining Rules, Proofs and Counterexamples 18 of 43 = df x isarectangle ∧ x isasquare all sidesof x are equal inlength .
Definition through a ruleforuse if and only if Df Greg Restall Defining Rules, Proofs and Counterexamples 19 of 43 [ X, A ⊗ B : Y ] is out of bounds [ X, A, B : Y ] is out of bounds
Definition through a ruleforuse Greg Restall Defining Rules, Proofs and Counterexamples 19 of 43 if and only if [ X, A ⊗ B : Y ] is out of bounds [ X, A, B : Y ] is out of bounds X, A, B � Y = = = = = = = = = = ⊗ Df X, A ⊗ B � Y
What about when to deny a conjunction? Id Df Cut Cut So, we have R Greg Restall Defining Rules, Proofs and Counterexamples 20 of 43 When do we have X � A ⊗ B, Y ?
What about when to deny a conjunction? Cut Defining Rules, Proofs and Counterexamples Greg Restall R So, we have Cut Id 20 of 43 When do we have X � A ⊗ B, Y ? A ⊗ B � A ⊗ B ⊗ Df X ′ � B, Y ′ A, B � A ⊗ B X ′ , A � A ⊗ B, Y ′ X � A, Y X, X ′ � A ⊗ B, Y, Y ′
What about when to deny a conjunction? Cut Defining Rules, Proofs and Counterexamples Greg Restall So, we have Cut Id 20 of 43 When do we have X � A ⊗ B, Y ? A ⊗ B � A ⊗ B ⊗ Df X ′ � B, Y ′ A, B � A ⊗ B X ′ , A � A ⊗ B, Y ′ X � A, Y X, X ′ � A ⊗ B, Y, Y ′ X ′ � B, Y ′ X � A, Y ⊗ R X, X ′ � A ⊗ B, Y, Y ′
What we've done in terms of norms governing simpler judgements. Greg Restall Defining Rules, Proofs and Counterexamples 21 of 43 We have given norms governing ⊗ judgements
Definitions for other logical concepts Df Defining Rules, Proofs and Counterexamples Greg Restall .) and are not present in and (Where Df Df Df Df 22 of 43 X � A, Y X, A � B, Y X � A, B, Y = = = = = = = = ¬ Df = = = = = = = = = = = → Df = = = = = = = = = = ⊕ Df X, ¬ A � Y X � A → B, Y X � A ⊕ B, Y
Definitions for other logical concepts Df Defining Rules, Proofs and Counterexamples Greg Restall .) and are not present in and (Where Df Df 22 of 43 X � A, Y X, A � B, Y X � A, B, Y = = = = = = = = ¬ Df = = = = = = = = = = = → Df = = = = = = = = = = ⊕ Df X, ¬ A � Y X � A → B, Y X � A ⊕ B, Y X � A, Y X � B, Y X, A � Y X, B � Y = = = = = = = = = = = = = = = ∧ Df = = = = = = = = = = = = = = = ∨ Df X � A ∧ B, Y X, A ∨ B � Y
Definitions for other logical concepts Greg Restall Defining Rules, Proofs and Counterexamples 22 of 43 X � A, Y X, A � B, Y X � A, B, Y = = = = = = = = ¬ Df = = = = = = = = = = = → Df = = = = = = = = = = ⊕ Df X, ¬ A � Y X � A → B, Y X � A ⊕ B, Y X � A, Y X � B, Y X, A � Y X, B � Y = = = = = = = = = = = = = = = ∧ Df = = = = = = = = = = = = = = = ∨ Df X � A ∧ B, Y X, A ∨ B � Y X � A | x X, A | x n , Y n � Y X, Fs � Ft, Y = = = = = = = = = = ∀ Df = = = = = = = = = = ∀ Df = = = = = = = = = = = Df X � ( ∀ x ) A, Y X, ( ∃ x ) A � Y X � s = t, Y (Where n and F are not present in X and Y .)
How does this work? How do concepts defined in this way work ? Greg Restall Defining Rules, Proofs and Counterexamples 23 of 43
Transforming Systems of Rules Df L/R L/R + Cut + Id L/R + Cut Id Elimination L/R + Cut L/R Cut Elimination Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43 ∗ Df + Cut + Id ⇔ ∗ L/R + Cut + Id
Transforming Systems of Rules L/R + Cut + Id L/R + Cut Id Elimination L/R + Cut L/R Cut Elimination Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43 ∗ Df + Cut + Id ⇔ ∗ L/R + Cut + Id ∗ Df ↔ ∗ L/R
Id Elimination Transforming Systems of Rules L/R + Cut L/R Cut Elimination Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43 ∗ Df + Cut + Id ⇔ ∗ L/R + Cut + Id ∗ Df ↔ ∗ L/R ∗ L/R + Cut + Id ⇔ ∗ L/R + Cut
Id Elimination Transforming Systems of Rules L/R + Cut L/R Cut Elimination Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43 ∗ Df + Cut + Id ⇔ ∗ L/R + Cut + Id ∗ Df ↔ ∗ L/R ∗ L/R + Cut + Id ⇔ ∗ L/R + Cut
Id Elimination Transforming Systems of Rules Cut Elimination Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43 ∗ Df + Cut + Id ⇔ ∗ L/R + Cut + Id ∗ Df ↔ ∗ L/R ∗ L/R + Cut + Id ⇔ ∗ L/R + Cut ∗ L/R + Cut ⇔ ∗ L/R
Id Elimination Transforming Systems of Rules Cut Elimination Greg Restall Defining Rules, Proofs and Counterexamples 24 of 43 ∗ Df + Cut + Id ⇔ ∗ L/R + Cut + Id ∗ Df ↔ ∗ L/R ∗ L/R + Cut + Id ⇔ ∗ L/R + Cut ∗ L/R + Cut ⇔ ∗ L/R
Concepts defined in this way… concept.) Are conservatively extending . (Adding a logical concept to your vocabulary in this way doesn’t constrain the bounds in the original language.) Play useful dialogical roles . (You can do things with these concepts that you cannot do without. Denying a conjunction does something different to denying the conjuncts.) Are subject matter neutral . (They work wherever you assert and deny—and have singular terms and predicates.) In Brandom’s terms, they make explicit some of what was implicit in the practice of assertion and denial. Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43 ▶ Are uniquely defined . (If you and I use the same rule, we define the same
Concepts defined in this way… concept.) in this way doesn’t constrain the bounds in the original language.) Play useful dialogical roles . (You can do things with these concepts that you cannot do without. Denying a conjunction does something different to denying the conjuncts.) Are subject matter neutral . (They work wherever you assert and deny—and have singular terms and predicates.) In Brandom’s terms, they make explicit some of what was implicit in the practice of assertion and denial. Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43 ▶ Are uniquely defined . (If you and I use the same rule, we define the same ▶ Are conservatively extending . (Adding a logical concept to your vocabulary
Concepts defined in this way… concept.) in this way doesn’t constrain the bounds in the original language.) cannot do without. Denying a conjunction does something different to denying the conjuncts.) Are subject matter neutral . (They work wherever you assert and deny—and have singular terms and predicates.) In Brandom’s terms, they make explicit some of what was implicit in the practice of assertion and denial. Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43 ▶ Are uniquely defined . (If you and I use the same rule, we define the same ▶ Are conservatively extending . (Adding a logical concept to your vocabulary ▶ Play useful dialogical roles . (You can do things with these concepts that you
Concepts defined in this way… concept.) in this way doesn’t constrain the bounds in the original language.) cannot do without. Denying a conjunction does something different to denying the conjuncts.) have singular terms and predicates.) In Brandom’s terms, they make explicit some of what was implicit in the practice of assertion and denial. Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43 ▶ Are uniquely defined . (If you and I use the same rule, we define the same ▶ Are conservatively extending . (Adding a logical concept to your vocabulary ▶ Play useful dialogical roles . (You can do things with these concepts that you ▶ Are subject matter neutral . (They work wherever you assert and deny—and
Concepts defined in this way… concept.) in this way doesn’t constrain the bounds in the original language.) cannot do without. Denying a conjunction does something different to denying the conjuncts.) have singular terms and predicates.) practice of assertion and denial. Greg Restall Defining Rules, Proofs and Counterexamples 25 of 43 ▶ Are uniquely defined . (If you and I use the same rule, we define the same ▶ Are conservatively extending . (Adding a logical concept to your vocabulary ▶ Play useful dialogical roles . (You can do things with these concepts that you ▶ Are subject matter neutral . (They work wherever you assert and deny—and ▶ In Brandom’s terms, they make explicit some of what was implicit in the
what proofs are & what they do
A Tiny Proof If it’s Thursday, I’m in Melbourne. It’s Thursday. Therefore, I’m in Melbourne. Id Df It’s Thursday I’m in Melbourne It’s Thursday I’m in Melbourne (This is out of bounds.) Greg Restall Defining Rules, Proofs and Counterexamples 27 of 43
A Tiny Proof If it’s Thursday, I’m in Melbourne. It’s Thursday. Therefore, I’m in Melbourne. Id It’s Thursday I’m in Melbourne It’s Thursday I’m in Melbourne (This is out of bounds.) Greg Restall Defining Rules, Proofs and Counterexamples 27 of 43 A → B � A → B → Df A → B, A � B
A Tiny Proof If it’s Thursday, I’m in Melbourne. It’s Thursday. Therefore, I’m in Melbourne. Id (This is out of bounds.) Greg Restall Defining Rules, Proofs and Counterexamples 27 of 43 A → B � A → B → Df A → B, A � B [ It’s Thursday → I’m in Melbourne , It’s Thursday : I’m in Melbourne ]
The stance ( pro or con ) The Undeniable Take a context in which I’ve asserted and I’ve asserted it’s Thursday , then I’m in Melbourne is undeniable . Adding the assertion makes explicit what was implicit before that assertion. on I’m in Melbourne was already made. Greg Restall Defining Rules, Proofs and Counterexamples 28 of 43 it’s Thursday → I’m in Melbourne
The stance ( pro or con ) The Undeniable Take a context in which I’ve asserted and I’ve asserted it’s Thursday , then I’m in Melbourne is undeniable . Adding the assertion makes explicit what was implicit before that assertion. on I’m in Melbourne was already made. Greg Restall Defining Rules, Proofs and Counterexamples 28 of 43 it’s Thursday → I’m in Melbourne
The Undeniable Take a context in which I’ve asserted and I’ve asserted it’s Thursday , then I’m in Melbourne is undeniable . Adding the assertion makes explicit what was implicit before that assertion. on I’m in Melbourne was already made. Greg Restall Defining Rules, Proofs and Counterexamples 28 of 43 it’s Thursday → I’m in Melbourne The stance ( pro or con )
Proofs by way of the defining rules for the concepts involved in the proof. In this sense, proofs are analytic . They apply, given the definitions, independently of the positions taken by those giving the proof. Greg Restall Defining Rules, Proofs and Counterexamples 29 of 43 A proof for X � Y shows that the position [ X : Y ] is out of bounds,
Proofs by way of the defining rules for the concepts involved in the proof. In this sense, proofs are analytic . They apply, given the definitions, independently of the positions taken by those giving the proof. Greg Restall Defining Rules, Proofs and Counterexamples 29 of 43 A proof for X � Y shows that the position [ X : Y ] is out of bounds,
What Proofs Prove and a refutation of from , and more . Greg Restall Defining Rules, Proofs and Counterexamples 30 of 43 A proof of A, B � C, D can be seen as a proof of C from [ A, B : D ] ,
What Proofs Prove and more . Greg Restall Defining Rules, Proofs and Counterexamples 30 of 43 A proof of A, B � C, D can be seen as a proof of C from [ A, B : D ] , and a refutation of A from [ B : C, D ] ,
counterexamples & kreisel’s squeeze
Enlarging Positions is also not derivable. Defining Rules, Proofs and Counterexamples Greg Restall or so is either is available, then If and then one of is not derivable If Cut 32 of 43 X � A, Y X, A � Y X � Y
Enlarging Positions Cut is also not derivable. If is available, then so is either or Greg Restall Defining Rules, Proofs and Counterexamples 32 of 43 X � A, Y X, A � Y X � Y If X � Y is not derivable then one of X, A � Y and X � A � Y
Enlarging Positions Cut is also not derivable. Greg Restall Defining Rules, Proofs and Counterexamples 32 of 43 X � A, Y X, A � Y X � Y If X � Y is not derivable then one of X, A � Y and X � A � Y If [ X : Y ] is available, then so is either [ X, A : Y ] or [ X : A, Y ]
Keep Going … we can extend it into of the entire language. is not derivable for any finite and . Greg Restall Defining Rules, Proofs and Counterexamples 33 of 43 If [ X : Y ] is available, a partition [ X ′ : Y ′ ]
Keep Going … we can extend it into of the entire language. Greg Restall Defining Rules, Proofs and Counterexamples 33 of 43 If [ X : Y ] is available, a partition [ X ′ : Y ′ ] U � V is not derivable for any finite U ⊆ X ′ and V ⊆ Y ′ .
Adding Witnesses and similarly when is added on the right. Df,W Df,W Greg Restall Defining Rules, Proofs and Counterexamples 34 of 43 If ( ∃ x ) A is added on the left, we also add a witness A | x n , where n is fresh
Adding Witnesses and similarly when is added on the right. Df,W Greg Restall Defining Rules, Proofs and Counterexamples 34 of 43 If ( ∃ x ) A is added on the left, we also add a witness A | x n , where n is fresh X, A | x n , ( ∃ x ) A � Y ∃ Df,W X, ( ∃ x ) A � Y
Adding Witnesses Greg Restall Defining Rules, Proofs and Counterexamples 34 of 43 If ( ∃ x ) A is added on the left, we also add a witness A | x n , where n is fresh and similarly when ( ∀ x ) A is added on the right. X, A | x X � ( ∀ x ) A, A | x n , ( ∃ x ) A � Y n , Y ∃ Df,W ∀ Df,W X, ( ∃ x ) A � Y X � ( ∀ x ) A, Y
Recommend
More recommend
